The problem of recovering planted community structure in random graphs has received a lot of attention in the literature on the stochastic block model, where the input is a random graph in which edges crossing between different communities appear with smaller probability than edges induced by communities. The communities themselves form a collection of vertex-disjoint sparse cuts in the expected graph, and can be recovered, often exactly, from a sample as long as a separation condition on the intra- and inter-community edge probabilities is satisfied. In this paper, we ask whether the presence of a large number of overlapping sparsest cuts in the expected graph still allows recovery. For example, the d-dimensional hypercube graph admits d distinct (balanced) sparsest cuts, one for every coordinate. Can these cuts be identified given a random sample of the edges of the hypercube where each edge is present independently with some probability p ∈ (0, 1)? We show that this is the case, in a very strong sense: the sparsest balanced cut in a sample of the hypercube at rate p = Clog d/d for a sufficiently large constant C is 1/poly(d)-close to a coordinate cut with high probability. This is asymptotically optimal and allows approximate recovery of all d cuts simultaneously. Furthermore, for an appropriate sample of hypercube-like graphs recovery can be made exact. The proof is essentially a strong hypercube cut sparsification bound that combines a theorem of Friedgut, Kalai and Naor on boolean functions whose Fourier transform concentrates on the first level of the Fourier spectrum with Karger’s cut counting argument.
Kapralov et al. (Thu,) studied this question.